Today we will prove a logarithmic identity. We need to show that c to the power of log base c of a times log base b of c equals a to the power of log base b of c. Our strategy is to start with the left-hand side and use exponent rules and logarithm properties to transform it into the right-hand side.
Let's start our proof by examining the left-hand side of the equation. We have c to the power of log base c of a times log base b of c. We can rewrite this using the exponent rule that x to the m to the n equals x to the mn. This allows us to rewrite our expression as c to the log base c of a, all raised to the power of log base b of c.
Now we apply the fundamental property of logarithms. This property states that c to the power of log base c of x equals x. In our case, c to the power of log base c of a simply equals a. Therefore, our expression becomes a to the power of log base b of c, which is exactly the right-hand side of our original equation.
Let's summarize our complete proof. We started with the original equation c to the power of log base c of a times log base b of c equals a to the power of log base b of c. In step one, we used the exponent rule to rewrite the left side as c to the log base c of a, all raised to the power of log base b of c. In step two, we applied the fundamental logarithm property to simplify c to the log base c of a to just a. This gave us a to the power of log base b of c, which is exactly our right-hand side. Therefore, we have proven that the left-hand side equals the right-hand side, completing our proof.
This logarithmic identity has many practical applications. It's useful for simplifying complex expressions, solving logarithmic equations, and working with change of base formulas. Let's verify our proof with a concrete example. If we let a equal 8, b equal 2, and c equal 2, then the left-hand side becomes 2 to the power of log base 2 of 8 times log base 2 of 2, which equals 2 to the 3 times 1, or 2 cubed, which is 8. The right-hand side is 8 to the power of log base 2 of 2, which is 8 to the first power, also equal to 8. This confirms our identity is correct.